{{Short description|Typical velocity of the thermal motion of particles}} '''Thermal velocity''' or '''thermal speed''' is a typical velocity of the thermal motion of particles that make up a gas, liquid, etc. Thus, indirectly, thermal velocity is a measure of temperature. Technically speaking, it is a measure of the width of the peak in the Maxwell–Boltzmann particle velocity distribution. Note that in the strictest sense ''thermal velocity'' is not a velocity, since ''velocity'' usually describes a vector rather than simply a scalar speed.

==Definitions==

Since the thermal velocity is only a "typical" velocity, a number of different definitions can be and are used.

Taking <math>k_\text{B}</math> to be the Boltzmann constant, <math>T</math> the absolute temperature, and <math>m</math> the mass of a particle, we can write the different thermal velocities:

=== In one dimension ===

If <math>v_\text{th}</math> is defined as the root mean square of the velocity in any one dimension (i.e. any single direction), then<ref name="Baumjohann">{{cite book | last = Baumjohann | first = Wolfgang | title = Basic Space Plasma Physics | last2 = Treumann | first2 = Rudolf A. | date = 2006 | publisher = Imperial College Press | isbn = 978-1-86094-079-8 | edition = Reprinted | location = London}}</ref><ref name="Gurnett">{{cite book | last = Gurnett | first = Donald A. | title = Introduction to Plasma Physics: With Space, Laboratory and Astrophysical Applications | last2 = Bhattacharjee | first2 = Amitava | date = 2017 | publisher = Cambridge University Press | isbn = 978-1-107-02737-4 | edition = 2nd | location = Cambridge}}</ref> <math display="block">v_\text{th} = \sqrt{\frac{k_\text{B} T}{m}}.</math>

If <math>v_\text{th}</math> is defined as the mean of the magnitude of the velocity in any one dimension (i.e. any single direction), then <math display="block">v_\text{th} = \sqrt{\frac{2 k_\text{B} T}{\pi m}}.</math>

=== In three dimensions ===

If <math>v_\text{th}</math> is defined as the most probable speed, then<ref name="Gurnett"/> <math display="block">v_\text{th} = \sqrt{\frac{2k_\text{B} T}{m}}.</math>

If <math>v_\text{th}</math> is defined as the root mean square of the total velocity, then <math display="block">v_\text{th} = \sqrt{\frac{3k_\text{B} T}{m}}.</math>

If <math>v_\text{th}</math> is defined as the mean of the magnitude of the velocity of the atoms or molecules, then <math display="block">v_\text{th} = \sqrt{\frac{8k_\text{B} T}{\pi m}}.</math>

All of these definitions are in the range <math display="block">v_\text{th} = (1.6 \pm 0.2) \sqrt{\frac{k_\text{B} T}{m}}.</math>

==Thermal velocity at room temperature==

At 20&nbsp;°C (293.15&nbsp;kelvins<!-- lower-case and pluralized! -->), the mean thermal velocity of common gasses in three dimensions is:<ref>{{Cite web |title=Thermal velocity |url=https://www.pfeiffer-vacuum.com/en/know-how/introduction-to-vacuum-technology/fundamentals/thermal-velocity/ |access-date=2023-05-28 |website=www.pfeiffer-vacuum.com |archive-url= https://web.archive.org/web/20230131163522/https://www.pfeiffer-vacuum.com/en/know-how/introduction-to-vacuum-technology/fundamentals/thermal-velocity/ |archive-date= 2023-01-31}}</ref> {| class="wikitable" |- ! Gas ! Thermal velocity |- | Hydrogen | {{convert|1,754|m/s|ft/s|abbr=on}} |- | Helium | {{convert|1,245|m/s|ft/s|abbr=on}} |- | Water vapor | {{convert|585|m/s|ft/s|abbr=on}} |- | Nitrogen | {{convert|470|m/s|ft/s|abbr=on}} |- | Air | {{convert|464|m/s|ft/s|abbr=on}} |- | Argon | {{convert|394|m/s|ft/s|abbr=on}} |- | Carbon dioxide | {{convert|375|m/s|ft/s|abbr=on}} |}

==References== {{Reflist}}

{{DEFAULTSORT:Thermal Velocity}} Category:Thermodynamic properties Category:Statistical mechanics

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